Sen. Mike Lee of Utah, a proponent of defunding, also said Thursday the fight to kill the healthcare law is not worth shutting the government down, which is what some Senate conservatives had threatened during Congress' summer recess.

"You saw this explode with anger publicly when they stood up . . . waving the white flag, and [said], 'Listen, we're not going to fight here. We're going to surrender. We can't win. We're going to send the hot potato back to the House, and they're going to fight there,'" Duffy said.

Cruz, however, said on Thursday that he plans to do everything he can to stop Obamacare, even if it means filibustering a bill to keep the government operating beyond Sept. 30, the end of the fiscal year.

Duffy said the stopgap spending measure passed by the House on Friday, complete with language defunding Obamacare, would give Cruz, Lee, and Florida Sen. Marco Rubio "exactly what they asked for — the opportunity to fight in the Senate on defunding Obamacare."

The congressman said passage of the House bill will force Cruz and his conservative cohorts to either put up or shut up.

"What I see happening now is, people coming out and calling them out for the hypocrisy of these big, tough conservatives who know how to fight, but will never get in the ring," he said, referring to a Cruz statement on Wednesday indicating the Democrats in the Senate would likely prevail in keeping Obamacare alive.

Duffy insisted that most congressional Republicans oppose a government shutdown, and he indicated that he sees room for negotiation over GOP objections to Obamacare funding that could prevent a government shutdown.

"America doesn't like Obamacare right now. They are confused by it. They don't want it. . . So we think we have an opportunity, maybe not to defund it, but we may get the president to agree to a delay in Obamacare," he said.

But the president, he said, has "come to the table" in order to "find that sweet spot of victory" that gives everyone at least some of what they want.